As the Obama administration scrambles to fix the failed health care enrollment website, Republicans are stepping up their attacks on the rollout.

HealthCare.gov will now be closed every day from 1 a.m. to 5 a.m. for repairs.

On NBC's "Meet the Press," former Republican presidential candidate Mitt Romney accused the president of being dishonest with the American people.

"I think the key thing that the president is trying to get away from is that he told people they could keep their insurance, and that was not the truth," he said. "And whether you like the model of Obamacare or not, the fact that the president sold it on a basis that was not true has undermined the foundation of his second term."

Romney said now the president needs to try to rebuild that foundation by actually working with Congress.

Some Democratic lawmakers are also frustrated with the many problems plaguing HealthCare.gov. They want the website to be completely shut down until it's fixed.

"You know, I felt -- and I said this directly to the president's chief of staff -- they ought to take down the website until it was right," Sen. Dianne Feinstein, D-Calif., said.

"They believe that they need to keep it running," she continued. "And that they can sort out the difficulties, that they brought in technological experts from a broad base of the private sector and that by the end of November, it can be sorted out and be functioning properly."

Sen. Kelly Ayotte, R-N.H., said, "I would say to the president -- why doesn't he call a timeout on this?"

There are also new questions about the security of putting everyone's information online. Rep. Mike Rogers, R-Mich., says the Obama administration has admitted the risk exists.

"They said that they think the risk was acceptable. But their information wasn't at risk, the American people's information was at risk," Rogers told CBS's "Face the Nation."

"The real threats to these systems when you have nation states, organized crime groups, and criminals trying to get information that is now available on these websites," he said.

Rogers said security needs to be stabilized before any other problems with the website are fixed.

Meanwhile, millions of Americans continue to get letters telling them their coverage has been cancelled.

In a column for the Wall Street Journal, a California woman suffering from gallbladder cancer called herself one of the "losers" of Obamacare.

She said Obamacare is forcing her to give up her world-class health plan, and she could lose her world-class doctor, too.

She ended by saying, "For a cancer patient, medical coverage is a matter of life and death. Take away people's ability to control their medical-coverage choices and they may die. I guess that's a highly effective way to control medical costs. Perhaps that's the point."

Meanwhile, President Barack Obama continues to push his healthcare plan.

On Wednesday, he travels to Dallas where he'll highlight the work being done in local communities across the country to prepare people for the full implementation of Obamacare.